Properties

Label 4-370e2-1.1-c1e2-0-9
Degree $4$
Conductor $136900$
Sign $1$
Analytic cond. $8.72886$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 5-s + 3·6-s − 8-s + 3·9-s + 10-s + 4·11-s + 13-s + 3·15-s − 16-s + 2·17-s + 3·18-s − 2·19-s + 4·22-s − 12·23-s − 3·24-s + 26-s + 12·29-s + 3·30-s + 10·31-s + 12·33-s + 2·34-s − 11·37-s − 2·38-s + 3·39-s − 40-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 0.447·5-s + 1.22·6-s − 0.353·8-s + 9-s + 0.316·10-s + 1.20·11-s + 0.277·13-s + 0.774·15-s − 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.458·19-s + 0.852·22-s − 2.50·23-s − 0.612·24-s + 0.196·26-s + 2.22·29-s + 0.547·30-s + 1.79·31-s + 2.08·33-s + 0.342·34-s − 1.80·37-s − 0.324·38-s + 0.480·39-s − 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(136900\)    =    \(2^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(8.72886\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{370} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 136900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.346043039\)
\(L(\frac12)\) \(\approx\) \(4.346043039\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
37$C_2$ \( 1 + 11 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.95163200795000568405637685859, −11.32321448255447855739248646614, −10.38904104770267075409135561590, −10.19166817942503132232718048092, −9.857731872583700502391303430650, −9.045073460082600879588754395068, −8.946564042132285791414141588271, −8.299301297591984611693436244611, −8.188868981264419101140333349812, −7.57527665202680956664320514258, −6.58063465243093294159014663586, −6.52228201100071469565806930147, −5.88171599656515378543321409274, −5.23746815707710114612601892896, −4.23807683064305712673795393728, −4.21201875262721930949205950758, −3.42054882189283528330028759162, −2.84862013964375338544521413445, −2.30393138872681614184910660274, −1.40283922902562266868125513243, 1.40283922902562266868125513243, 2.30393138872681614184910660274, 2.84862013964375338544521413445, 3.42054882189283528330028759162, 4.21201875262721930949205950758, 4.23807683064305712673795393728, 5.23746815707710114612601892896, 5.88171599656515378543321409274, 6.52228201100071469565806930147, 6.58063465243093294159014663586, 7.57527665202680956664320514258, 8.188868981264419101140333349812, 8.299301297591984611693436244611, 8.946564042132285791414141588271, 9.045073460082600879588754395068, 9.857731872583700502391303430650, 10.19166817942503132232718048092, 10.38904104770267075409135561590, 11.32321448255447855739248646614, 11.95163200795000568405637685859

Graph of the $Z$-function along the critical line